On soft submaximal spaces

In the present paper, detailed properties of soft submaximal topological spaces have been discussed. It starts by examining how well soft submaximal spaces behave when subjected to some operations, such as: taking soft subspaces, soft products, soft topological sum and images/preimages under certain soft mappings. Furthermore, several characterizations of soft submaximal spaces are obtained with respect to various types of soft subsets of soft topological spaces and their simple extensions.


Introduction
The area of topology which focuses on the fundamental set-theoretic concepts and procedures utilized in topology is known as general topology. It is the foundation of other branches of topology, including geometric topology, algebraic topology, and differential topology. The notion of submaximal spaces was introduced by Bourbaki [15] as a tool to study the topological spaces that do not admit a larger topology with the same semi-regularization. Such spaces were considered as a significant topic in both topological space and topological group, (see [10,17]). Soft topology, particularly merges soft set theory and topology, is also a field of topology. It is driven by the basic assumptions of classic topological space and focuses on the set of all soft sets. A parameterized family of sets is known as a soft set. It was proposed by Molodtsov [28] in 1999 to concern with uncertainties. Shabir and Naz's [30] contributions were especially important in establishing the area of soft topology. Later several subclasses of soft topological spaces were suggested, including soft separation axioms [1,3,14], soft separable spaces [14], soft connected [25], soft compact [13,32], soft paracompact [25], soft extremally disconnected [11], soft J-spaces [27], and soft (nearly) Menger spaces [4,24]. Furthermore, soft bioperators on soft topological spaces have been studied in [12]. Despite the fact that many studies followed their instructions and many notions developed in soft environments, substantial contributions can also be provided. The class of soft submaximal spaces was defined by Ilango and Ravindran [21], but not much study was done to further investigate soft submaximal spaces. The topic is crucial for considering the maximal element in the set of all soft topologies (see [2]) or soft topological groups on a * Corresponding author.
E-mail address: algore@just.edu.jo (S. Al Ghour). common universe, and for extending many other different results done in [10]. Hence, we develop the properties of soft submaximal spaces and obtain several of their characterizations.
If both mappings and ℎ are bijective, then is bijective.
Definition 2.9. [29] A soft mapping from a soft topological space We start by showing soft submaximality is a hereditary property.
Also, the other parts can be followed from (1) and (2). □

Conclusion
After the work of Shabir and Naz [30], several types of soft topological spaces have been analyzed. For example: soft compact, soft connected, soft paracompact, soft extremally disconnected, soft separable, soft -spaces, = 0, 1, … , 4, and so on. We have continued working in the same direction by studying the class of soft submaximal spaces. We have shown any soft subspace of a soft submaximal space is soft submaximal. The soft product of two soft submaximal spaces need not be soft submaximal. The soft sum of any collection of soft submaximal spaces is soft submaximal space. The soft submaximal spaces are preserved under soft open surjections. Furthermore, we have described soft submaximal spaces in terms of different forms of soft sets.

Author contribution statement
Samer Al Ghour, Zanyar A. Ameen: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.